Subgroups — #30839843

Order 17152.

766 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569669	C₂	1	2	yes	maximal, complemented
2	C₂×#16863475	C₂	1	1	yes	maximal, characteristic, complemented
3	#22559767	C₂	1	4	yes	maximal, complemented
4	#22559719	C₂	1	2	yes	maximal, complemented
5	#22559685	C₂	1	2	yes	maximal, complemented
6	#22558965	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558886	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3169	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14524068	—	2	2	no	—
10	C₂²×#11761611	—	2	1	no	—
11	C₂×C₄×D₂₆₈	C₂²	1	1	yes	characteristic
12	#16863608	—	2	4	no	—
13	#16863598	C₂²	1	4	yes	—
14	#16863588	C₂²	1	4	yes	—
15	#16863475	—	2	1	no	—
16	#16863475	—	2	1	no	—
17	#16863475	C₂²	1	2	yes	—
18	#16863475	C₂²	1	2	yes	—
19	#16863420	C₂²	1	2	yes	—
20	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
21	#16863397	—	2	2	no	—
22	C₂×(C₂×D₆₇)⋊C₈	—	2	1	no	—
23	#16863147	C₂²	1	2	yes	complemented
24	#16863147	C₂²	1	2	yes	complemented
25	#16863142	C₂²	1	4	yes	complemented
26	#16863140	C₂²	1	4	yes	complemented
27	#532×C₆₇	C₂²	1	2	yes	—
28	#500×C₆₇	C₂²	1	1	yes	characteristic, complemented
29	#493×C₆₇	C₂²	1	1	yes	characteristic, complemented
30	#486×C₆₇	C₂²	1	2	yes	complemented
31	#466×C₆₇	—	2	1	no	—
32	#466×C₆₇	—	2	1	no	—
33	C₄×D₈×C₆₇	—	2	1	no	—
34	C₄×D₈×C₆₇	—	2	1	no	—
35	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
36	C₂²×D₂₆₈	D₄	1	1	yes	characteristic
37	C₂²×C₄×D₆₇	—	2	1	no	—
38	#14524068	—	4	8	no	—
39	C₂×#11761615	—	2	4	no	—
40	C₂×#11761613	—	2	4	no	—
41	C₂×#11761612	—	2	2	no	—
42	C₂×#11761611	—	2	2	no	—
43	C₂×#11761611	—	2	2	no	—
44	C₂×#11761611	—	4	4	no	—
45	C₂²×C₆₇⋊C₈	—	2	1	no	—
46	C₄×D₂₆₈	—	2	1	no	—
47	C₄×D₂₆₈	—	2	1	no	—
48	C₄×D₂₆₈	D₄	1	2	yes	—
49	C₄×D₂₆₈	C₂³	1	2	yes	—
50	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
51	C₂×(C₂×D₆₇)⋊C₄	—	2	1	no	—
52	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
53	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
54	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
55	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
56	#14523967	—	2	4	no	—
57	#14523964	—	2	4	no	—
58	(C₂×D₆₇)⋊C₈	—	2	2	no	—
59	(C₂×D₆₇)⋊C₈	—	2	2	no	—
60	C₂×#41×C₆₇	—	2	2	no	—
61	C₂×D₈×C₆₇	—	2	1	no	—
62	C₂×D₈×C₆₇	—	2	1	no	—
63	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
65	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
66	#127×C₆₇	C₂³	1	2	yes	—
67	#124×C₆₇	—	2	2	no	—
68	#121×C₆₇	—	2	2	no	—
69	#121×C₆₇	—	2	2	no	—
70	C₄×D₄×C₆₇	—	2	2	no	—
71	C₄×D₄×C₆₇	—	2	2	no	—
72	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
73	#107×C₆₇	—	2	1	no	—
74	#107×C₆₇	—	2	1	no	—
75	#102×C₆₇	—	2	2	no	—
76	#102×C₆₇	—	2	2	no	—
77	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
78	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
79	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
81	C₂³×D₆₇	—	2	1	no	—
82	C₂×D₂₆₈	—	2	4	no	—
83	C₂×D₂₆₈	—	2	1	no	—
84	C₂×D₂₆₈	C₂×D₄	1	2	yes	—
85	C₂²×Dic₆₇	—	2	1	no	—
86	C₂×C₄×D₆₇	—	2	2	no	—
87	C₂×C₄×D₆₇	—	2	2	no	—
88	C₂×C₄×D₆₇	—	4	2	no	—
89	C₂×C₄×D₆₇	—	4	2	no	—
90	#11761615	—	4	8	no	—
91	#11761613	—	4	8	no	—
92	#11761612	—	4	4	no	—
93	#11761611	—	4	4	no	—
94	#11761611	—	4	4	no	—
95	C₂×C₆₇⋊C₈	—	2	2	no	—
96	C₂×C₆₇⋊C₈	—	2	4	no	—
97	C₂×C₆₇⋊C₈	—	4	2	no	—
98	C₄⋊Dic₆₇	—	2	1	no	—
99	C₄⋊Dic₆₇	C₂×D₄	1	2	yes	—
100	(C₂×D₆₇)⋊C₄	—	2	2	no	—
101	(C₂×D₆₇)⋊C₄	—	2	2	no	—
102	#41×C₆₇	—	4	4	no	—
103	#41×C₆₇	—	4	4	no	—
104	C₂×Q₈×C₆₇	—	2	2	no	—
105	C₂×D₄×C₆₇	—	2	2	no	—
106	C₂×D₄×C₆₇	—	2	2	no	—
107	C₂×D₄×C₆₇	—	2	2	no	—
108	C₂×D₄×C₆₇	—	4	2	no	—
109	C₂²×C₄×C₆₇	—	4	2	no	—
110	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
113	D₈×C₆₇	—	4	2	no	—
114	D₈×C₆₇	—	4	2	no	—
115	C₂×C₈×C₆₇	—	2	2	no	—
116	C₂×C₈×C₆₇	—	4	1	no	—
117	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₄⋊C₄×C₆₇	—	2	2	no	—
122	C₄⋊C₄×C₆₇	—	2	2	no	—
123	C₂²⋊C₄×C₆₇	—	4	2	no	—
124	C₂²⋊C₄×C₆₇	—	4	2	no	—
125	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₆₇	#41	1	1	yes	characteristic
128	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
129	C₂³×C₆₇	—	4	2	no	—
130	C₂³×C₆₇	#121	1	1	yes	characteristic
131	C₂²×D₆₇	—	2	2	no	—
132	C₂²×D₆₇	—	4	2	no	—
133	C₂²×D₆₇	—	4	2	no	—
134	C₂²×D₆₇	—	4	2	no	—
135	Q₈×C₆₇	—	4	2	no	—
136	Q₈×C₆₇	—	4	2	no	—
137	D₄×C₆₇	—	4	2	no	—
138	D₄×C₆₇	—	4	2	no	—
139	D₄×C₆₇	—	4	2	no	—
140	D₄×C₆₇	—	4	2	no	—
141	D₄×C₆₇	—	8	2	no	—
142	D₄×C₆₇	—	8	4	no	—
143	C₂×C₄×C₆₇	—	2	1	no	—
144	C₂×C₄×C₆₇	—	2	1	no	—
145	C₂×C₄×C₆₇	—	4	1	no	—
146	C₂×C₄×C₆₇	—	4	2	no	—
147	C₂×C₄×C₆₇	—	8	4	no	—
148	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
149	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
150	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
151	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
152	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
155	C₂×Dic₆₇	—	2	2	no	—
156	C₂×Dic₆₇	—	4	2	no	—
157	D₂₆₈	—	2	4	no	—
158	D₂₆₈	—	4	2	no	—
159	C₄×D₆₇	—	4	4	no	—
160	C₄×D₆₇	—	4	2	no	—
161	C₄×D₆₇	—	4	2	no	—
162	C₈×C₆₇	—	2	1	no	—
163	C₈×C₆₇	—	2	1	no	—
164	C₈×C₆₇	—	2	1	no	—
165	C₈×C₆₇	AGL(1,ℤ/8)	1	2	yes	complemented
166	C₆₇⋊C₈	—	4	2	no	—
167	C₆₇⋊C₈	—	4	4	no	—
168	C₂²×C₆₇	—	2	1	no	—
169	C₂²×C₆₇	—	2	1	no	—
170	C₂²×C₆₇	—	8	2	no	—
171	C₂²×C₆₇	—	8	2	no	—
172	C₂²×C₆₇	—	8	2	no	—
173	C₂²×C₆₇	#469	1	2	yes	—
174	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
175	C₂×D₆₇	—	4	2	no	—
176	C₂×D₆₇	—	4	2	no	—
177	C₂×D₆₇	—	4	2	no	—
178	C₂×D₆₇	—	8	4	no	—
179	C₄×C₆₇	—	2	1	no	—
180	C₄×C₆₇	—	2	1	no	—
181	C₄×C₆₇	—	2	1	no	—
182	C₄×C₆₇	—	8	2	no	—
183	C₄×C₆₇	#469	1	2	yes	—
184	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
185	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
186	C₄×C₆₇	#575	1	1	yes	characteristic
187	Dic₆₇	—	4	2	no	—
188	#19280	—	67	1	no	maximal, Sylow
189	C₂×C₆₇	—	2	2	no	—
190	C₂×C₆₇	—	16	2	no	—
191	C₂×C₆₇	#3036	1	1	yes	characteristic
192	C₂×C₆₇	C₂×#469	1	1	yes	characteristic
193	C₂×C₆₇	#3067	1	1	yes	characteristic
194	D₆₇	—	8	2	no	—
195	#3169	D₆₇	1	1	yes	characteristic, complemented
196	#3036	—	67	2	no	—
197	C₂×#436	—	67	1	no	—
198	#1701	—	67	4	no	—
199	#1693	—	67	2	no	—
200	#1687	—	67	2	no	—
201	#1566	—	67	1	no	—
202	#1547	—	67	2	no	—
203	C₆₇	#19280	1	1	yes	characteristic, complemented, Sylow
204	C₂×AGL(1,ℤ/8)	—	134	2	no	—
205	C₂²×M₁₆	—	134	1	no	—
206	#532	C₂×D₆₇	1	2	yes	—
207	C₂×C₄×D₄	—	67	1	no	—
208	#500	C₂×D₆₇	1	1	yes	characteristic, complemented
209	#493	C₂×D₆₇	1	1	yes	characteristic, complemented
210	#486	C₂×D₆₇	1	2	yes	complemented
211	#466	—	2	1	no	—
212	#466	—	2	1	no	—
213	#463	—	134	4	no	—
214	#461	—	67	4	no	—
215	#459	—	67	4	no	—
216	C₄×D₈	—	2	1	no	—
217	C₄×D₈	—	2	1	no	—
218	#436	—	67	2	no	—
219	#436	—	67	2	no	—
220	#436	—	134	1	no	—
221	#436	—	134	1	no	—
222	#423	—	67	2	no	—
223	C₂×C₄⋊C₈	—	67	1	no	—
224	#417	—	134	2	no	—
225	C₂×C₂²⋊C₈	—	134	1	no	—
226	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
227	#334	—	67	2	no	—
228	#334	—	67	2	no	—
229	#332	—	67	4	no	—
230	#331	—	67	4	no	—
231	C₂×#41	—	2	2	no	—
232	C₂²×D₄	—	67	1	no	—
233	C₂³×C₄	—	134	1	no	—
234	AGL(1,ℤ/8)	—	268	8	no	—
235	C₂×SD₁₆	—	134	4	no	—
236	C₂×D₈	—	2	1	no	—
237	C₂×D₈	—	2	1	no	—
238	C₂×D₈	—	134	4	no	—
239	C₂×M₁₆	—	268	4	no	—
240	C₂×M₁₆	—	134	2	no	—
241	C₂×M₁₆	—	134	2	no	—
242	C₂×M₁₆	—	134	2	no	—
243	C₂²×C₈	—	134	1	no	—
244	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
245	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
246	C₄⋊Q₈	C₂²×D₆₇	1	2	yes	—
247	#127	C₂²×D₆₇	1	2	yes	—
248	#124	—	2	2	no	—
249	#121	—	2	2	no	—
250	#121	—	2	2	no	—
251	C₄×D₄	—	2	2	no	—
252	C₄×D₄	—	2	2	no	—
253	C₄×D₄	—	67	2	no	—
254	C₄×D₄	—	67	2	no	—
255	C₄×D₄	—	134	1	no	—
256	C₄×D₄	—	134	1	no	—
257	C₂×C₄⋊C₄	—	67	1	no	—
258	C₂×C₂²⋊C₄	—	134	1	no	—
259	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
260	#107	—	2	1	no	—
261	#107	—	2	1	no	—
262	C₄⋊C₈	—	67	2	no	—
263	C₄⋊C₈	—	67	4	no	—
264	C₄⋊C₈	—	134	2	no	—
265	C₄⋊C₈	—	134	1	no	—
266	#103	—	134	4	no	—
267	#102	—	2	2	no	—
268	#102	—	2	2	no	—
269	#102	—	134	4	no	—
270	C₂²⋊C₈	—	134	2	no	—
271	C₂²⋊C₈	—	134	2	no	—
272	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
273	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
274	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
275	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
276	C₂⁴	—	134	1	no	—
277	#41	—	4	4	no	—
278	#41	—	4	4	no	—
279	C₂×Q₈	—	2	2	no	—
280	C₂×D₄	—	2	2	no	—
281	C₂×D₄	—	2	2	no	—
282	C₂×D₄	—	2	2	no	—
283	C₂×D₄	—	4	2	no	—
284	C₂×D₄	—	67	2	no	—
285	C₂×D₄	—	134	1	no	—
286	C₂×D₄	—	134	4	no	—
287	C₂²×C₄	—	4	2	no	—
288	C₂²×C₄	—	268	2	no	—
289	C₂²×C₄	—	268	2	no	—
290	C₂²×C₄	—	134	1	no	—
291	C₂²×C₄	—	134	2	no	—
292	C₂²×C₄	—	134	2	no	—
293	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
294	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
295	C₂²×C₄	#11761627	1	1	yes	characteristic
296	SD₁₆	—	268	8	no	—
297	D₈	—	4	2	no	—
298	D₈	—	4	2	no	—
299	D₈	—	268	8	no	—
300	M₁₆	—	268	4	no	—
301	M₁₆	—	268	4	no	—
302	M₁₆	—	268	4	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	4	1	no	—
305	C₂×C₈	—	268	2	no	—
306	C₂×C₈	—	134	4	no	—
307	C₂×C₈	—	134	2	no	—
308	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
309	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
310	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
311	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	67	2	no	—
315	C₄⋊C₄	—	134	1	no	—
316	C₂²⋊C₄	—	4	2	no	—
317	C₂²⋊C₄	—	4	2	no	—
318	C₂²⋊C₄	—	134	2	no	—
319	C₂²⋊C₄	—	134	2	no	—
320	C₄²	D₄×D₆₇	1	1	yes	characteristic
321	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
322	C₄²	#11761627	1	1	yes	characteristic
323	C₄²	C₂³×D₆₇	1	1	yes	characteristic
324	C₂³	—	4	2	no	—
325	C₂³	—	268	2	no	—
326	C₂³	—	268	2	no	—
327	C₂³	—	268	2	no	—
328	C₂³	—	134	2	no	—
329	C₂³	#14524014	1	1	yes	characteristic
330	Q₈	—	4	2	no	—
331	Q₈	—	4	2	no	—
332	D₄	—	4	2	no	—
333	D₄	—	4	2	no	—
334	D₄	—	4	2	no	—
335	D₄	—	4	2	no	—
336	D₄	—	8	2	no	—
337	D₄	—	8	4	no	—
338	D₄	—	268	2	no	—
339	D₄	—	134	4	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	2	no	—
344	C₂×C₄	—	8	4	no	—
345	C₂×C₄	—	268	4	no	—
346	C₂×C₄	—	268	2	no	—
347	C₂×C₄	—	268	2	no	—
348	C₂×C₄	—	268	2	no	—
349	C₂×C₄	—	134	2	no	—
350	C₂×C₄	#14524014	1	1	yes	characteristic
351	C₂×C₄	#14524014	1	1	yes	characteristic
352	C₂×C₄	#14524014	1	1	yes	characteristic
353	C₂×C₄	#14524067	1	4	yes	—
354	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
355	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
356	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
357	C₈	—	2	1	no	—
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	268	4	no	—
361	C₈	—	268	2	no	—
362	C₈	#14524068	1	2	yes	complemented
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	8	2	no	—
366	C₂²	—	8	2	no	—
367	C₂²	—	8	2	no	—
368	C₂²	—	268	2	no	—
369	C₂²	—	268	2	no	—
370	C₂²	—	268	2	no	—
371	C₂²	—	536	4	no	—
372	C₂²	#16863646	1	2	yes	—
373	C₂²	C₂×#14524014	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	8	2	no	—
378	C₄	—	268	2	no	—
379	C₄	#16863646	1	2	yes	—
380	C₄	C₂×#14524067	1	2	yes	—
381	C₄	C₂×#14524068	1	1	yes	characteristic
382	C₄	#16864194	1	1	yes	characteristic
383	C₂	—	2	2	no	—
384	C₂	—	16	2	no	—
385	C₂	—	536	2	no	—
386	C₂	#22569669	1	1	yes	characteristic
387	C₂	C₂×#16863646	1	1	yes	characteristic
388	C₂	#22569925	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14524014
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#3169	C₂
socle	C₂²×C₆₇	C₂×#121
hypercenter	#3169	D₆₇
hypocenter	C₆₇	#19280
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3169

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #22569669	C₂	semidirect
2. C₂×#16863475	C₂	semidirect
3. #22559767	C₂	semidirect
4. #22559719	C₂	semidirect
5. #22559685	C₂	semidirect
6. #22558965	C₂	semidirect
7. #22558886	C₂	semidirect
8. C₆₇×#3169	C₂	semidirect
23. #16863147	C₂²	semidirect
24. #16863147	C₂²	semidirect
25. #16863142	C₂²	semidirect
26. #16863140	C₂²	semidirect
28. #500×C₆₇	C₂²	semidirect
29. #493×C₆₇	C₂²	semidirect
30. #486×C₆₇	C₂²	semidirect
165. C₈×C₆₇	AGL(1,ℤ/8)	semidirect
195. #3169	D₆₇	semidirect
203. C₆₇	#19280	semidirect
208. #500	C₂×D₆₇	semidirect
209. #493	C₂×D₆₇	semidirect
210. #486	C₂×D₆₇	semidirect
362. C₈	#14524068	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22569669	2
C₂×#16863475	2
#22559767	2
#22559719	2
#22559685	2
#22558965	2
#22558886	2
C₆₇×#3169	2
#19280	67